Call or visit churches or synagogues in your area. If you are already a member of a church or religious organization, ask the church secretary or congregation members if an accompanist is needed. Offer your services for back-up if no regular positions are available---when the current accompanist is ill, for example, or for holidays and special services.
Call or visit schools in the area. Most employ full-time music teachers, but it can't hurt to pay a visit and drop off a business card. Ask about opportunities for seasonal plays, productions and concerts, and for after-school programs.
Call the music department of a local university or community college, and ask questions about available opportunities. The department or instructors may keep a list of accompanists. Offer to perform an audition, and ask if there are any special requirements for accompanists. Colleges and universities are often affiliated with civic bands, orchestras, ensembles and musicians, so ask for names and phone numbers of anyone who may be able to use your services. Check the bulletin boards in the music department, and post a flyer.
Go online. If your area has a local job board, look through the postings. Check other websites like Craigslist; if nothing is available, post an ad under "services," including the times and days of the week you are available. Also check the website of your local newspaper; classified ads are often posted online.
Become involved. As a musician, opportunities for both accompanying and teaching can open up simply from being involved in a band, orchestra, chorus or other music-related group. Even if you are not much of a singer, find a chorus that you can join to meet other musicians. Performing vocalists are likely to need an accompanist at some point, and it can't hurt to make acquaintances and let your talent be known.
Find out whether your area offers a Musicians' Guild. Services typically extended to dues-paying members include access to job listings, paid overtime hours and other services geared toward improving working conditions. For example, the Quebec Musicians' Guild, according to former president Gérard Masse, is "a union recognized by the laws governing trade unions [that is] certified as the sole representative for musicians in Quebec."